392 PROCEEDINGS OF THE AMERICAN ACADEMY § 9 



the volume, from 0° (or else the melting point) up to the 

 boiling point in each liquid, has been calculated both ac- 

 cording to Pierre and according to Kopp. These values 

 will be found in Tables VII., in the columns headed Vp 

 and Vj^ respectively. 



The mean was then taken in the column Fp a-, and the 

 difference between Kopp and Pierre, in the column head- 

 ed Ap_A', is squared in the next column, A^. 



With the method of least squares, by actual trial, the 

 best value of e in Table III. was found, to represent the 

 volume through the same range of temperature. The 

 value of e is given at the head of the table, and the 

 volumes, taken from Table III. by interpolation, are in 

 the column V (. The differences between these and the 

 mean volumes, according to Kopp and Pierre, are given in 

 the column A/>, ^r-t? ^^id the squares are given in the 

 last column, A^ 



These columns of squares are added for each liquid, so 

 that the sum of all the squares may easily be found. 



Amongst this number, or amongst the seventy-five previ- 

 ously examined, if there be a single liquid (as for instance 

 butyric acid) in which a molecular change affecting the 

 volume is brought about by heat, the mean difference 

 between theory and observation will be indefinitely in- 

 creased; if there be the slightest constant error in the ob- 

 servation, the elimination of which is impossible, the 

 observations will be equally accordant, but the theory will 

 seem unduly to disagree; in all cases, the mean square of 

 the difference between theory and observation will be in- 

 creased by the sum of the mean squares of the errors from 

 each source. 



There is no reason to suppose that the constants of the 

 standards, in terms of which the expansion was expressed, 

 as, for instance, the coefficients of expansion of mercury 

 and glass, determined by Regnault, were more accurately 



